Elimination Distances, Blocking Sets, and Kernels for Vertex Cover

The Vertex Cover problem plays an essential role in the study of polynomial kernelization in parameterized complexity, i.e., the study of provable and efficient preprocessing for NP-hard problems. Motivated by the great variety of positive and negative results for kernelization for Vertex Cover subject to different parameters and graph classes, we seek to unify and generalize them using so-called blocking sets, which have played implicit and explicit roles in many results. We show that in the most-studied setting, parameterized by the size of a deletion set to a specified graph class $\mathcal{C}$, bounded minimal blocking set size is necessary but not sufficient to get a polynomial kernelization. Under mild technical assumptions, bounded minimal blocking set size is showed to allow an essentially tight efficient reduction in the number of connected components. We then determine the exact maximum size of minimal blocking sets for graphs of bounded elimination distance to any hereditary class $\mathcal{C}$, including the case of graphs of bounded treedepth. We get similar but not tight bounds for certain non-hereditary classes $\mathcal{C}$, including the class $\mathcal{C}{LP}$ of graphs where integral and fractional vertex cover size coincide. These bounds allow us to derive polynomial kernels for Vertex Cover parameterized by the size of a deletion set to graphs of bounded elimination distance to, e.g., forest, bipartite, or $\mathcal{C}{LP}$ graphs.

💡 Research Summary

The paper investigates the Vertex Cover problem from the perspective of kernelization, focusing on two central notions: blocking sets and elimination distance. A blocking set Y⊆V(G) is a vertex set that cannot be contained in any optimal vertex cover of G; a minimal blocking set is inclusion‑minimal with this property. The authors first establish that for any graph class C closed under disjoint union (or, more strongly, robust), the existence of a graph in C with a minimal blocking set of size d implies that Vertex Cover parameterized by the size of a C‑modulator cannot admit a kernel of size O(|X|^{d−ε}) for any ε>0 unless NP⊆coNP/poly. This yields a unified lower‑bound framework that captures all known kernel lower bounds for deletion‑distance parameters, such as outerplanar graphs, mock forests, or any class containing all cliques.

However, bounded minimal blocking set size alone is not sufficient for a polynomial kernel. The authors construct a class C where every graph has minimal blocking sets of size one, yet Vertex Cover on C remains NP‑hard (indeed not even in RP unless NP=RP). Thus, additional algorithmic properties of C are required.

The paper then introduces elimination distance ed_C(G), a generalization of treedepth: it is the minimum number of vertex deletions needed to reduce G to a graph belonging to C, allowing the process to stop once each component lies in C. For hereditary, robust classes C with bounded β_C (the maximum size of a minimal blocking set in C), the authors precisely determine the maximum size β_C(d) of a minimal blocking set in graphs with elimination distance at most d to C. The main formula (Theorem 4) is:

• If β_C=1, then β_C(d)=2^{d−1}+1.
• If β_C≥2, then β_C(d)=(β_C−1)·2^{d}+1. These bounds are tight; the lower bound holds even for non‑hereditary classes, while the upper bound for non‑hereditary classes requires an additional f‑robustness condition. As a corollary, for treedepth d (i.e., elimination distance to the class of edgeless graphs), the exact bound is β(d)=1 for d=1 and β(d)=2^{d−2}+1 for d≥2, matching known results.

Armed with these bounds, the authors design a generic kernelization step: given an instance (G,k,X) where X is a C‑modulator and C is hereditary with polynomial‑time recognisable blocking sets, one can reduce the number of connected components of G−X to O(|X|·β_C) in polynomial time (Theorem 3). This component‑reduction is essentially optimal, as any improvement would contradict the lower‑bound of Theorem 1. After this reduction, standard techniques (e.g., bounding component size, applying known kernels for the base class) yield a full polynomial kernel.

Finally, the paper applies the framework to several concrete classes. For forests, bipartite graphs, and the class C_{LP} (graphs where the integer and fractional vertex‑cover numbers coincide), existing polynomial kernels for deletion‑distance parameters are lifted to kernels for deletion‑distance to graphs of bounded elimination distance to these classes. This subsumes earlier results for forests (parameterized by treedepth) and bipartite graphs, and introduces new kernels for the LP‑class. The results demonstrate that the combination of bounded blocking‑set size and bounded elimination distance provides a powerful, unified method for obtaining polynomial kernels for Vertex Cover across a wide spectrum of structural parameters.